LMFDB - Embedded newform 1104.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,4,Mod(1,1104)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1104.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1104.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$65.1381086463$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -10.7639 q^{5} -10.6525 q^{7} +9.00000 q^{9} +52.3607 q^{11} -79.0820 q^{13} +32.2918 q^{15} -77.2361 q^{17} -50.7902 q^{19} +31.9574 q^{21} +23.0000 q^{23} -9.13777 q^{25} -27.0000 q^{27} +12.7477 q^{29} +12.4133 q^{31} -157.082 q^{33} +114.663 q^{35} -73.1084 q^{37} +237.246 q^{39} -38.8916 q^{41} -171.787 q^{43} -96.8754 q^{45} -614.545 q^{47} -229.525 q^{49} +231.708 q^{51} +269.597 q^{53} -563.607 q^{55} +152.371 q^{57} +534.768 q^{59} -838.604 q^{61} -95.8723 q^{63} +851.234 q^{65} +448.180 q^{67} -69.0000 q^{69} -628.604 q^{71} +925.266 q^{73} +27.4133 q^{75} -557.771 q^{77} +963.479 q^{79} +81.0000 q^{81} -133.358 q^{83} +831.364 q^{85} -38.2430 q^{87} -778.581 q^{89} +842.420 q^{91} -37.2399 q^{93} +546.703 q^{95} +1603.57 q^{97} +471.246 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 26 q^{5} + 10 q^{7} + 18 q^{9} + 60 q^{11} - 24 q^{13} + 78 q^{15} - 150 q^{17} + 46 q^{19} - 30 q^{21} + 46 q^{23} + 98 q^{25} - 54 q^{27} - 216 q^{29} - 324 q^{31} - 180 q^{33} - 200 q^{35}+ \cdots + 540 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 26 * q^5 + 10 * q^7 + 18 * q^9 + 60 * q^11 - 24 * q^13 + 78 * q^15 - 150 * q^17 + 46 * q^19 - 30 * q^21 + 46 * q^23 + 98 * q^25 - 54 * q^27 - 216 * q^29 - 324 * q^31 - 180 * q^33 - 200 * q^35 + 140 * q^37 + 72 * q^39 - 364 * q^41 + 126 * q^43 - 234 * q^45 - 120 * q^47 - 146 * q^49 + 450 * q^51 + 490 * q^53 - 680 * q^55 - 138 * q^57 + 32 * q^59 - 908 * q^61 + 90 * q^63 + 12 * q^65 + 874 * q^67 - 138 * q^69 - 488 * q^71 + 652 * q^73 - 294 * q^75 - 400 * q^77 + 1198 * q^79 + 162 * q^81 + 100 * q^83 + 1940 * q^85 + 648 * q^87 + 102 * q^89 + 1980 * q^91 + 972 * q^93 - 928 * q^95 + 1624 * q^97 + 540 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−10.7639	−0.962755	−0.481378	−	0.876513i	$-0.659863\pi$
$5$	−10.7639	−0.962755	−0.481378	+	0.876513i	$0.659863\pi$
$6$	0	0
$7$	−10.6525	−0.575180	−0.287590	−	0.957754i	$-0.592854\pi$
$7$	−10.6525	−0.575180	−0.287590	+	0.957754i	$0.592854\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	52.3607	1.43521	0.717606	−	0.696449i	$-0.245238\pi$
$11$	52.3607	1.43521	0.717606	+	0.696449i	$0.245238\pi$
$12$	0	0
$13$	−79.0820	−1.68719	−0.843593	−	0.536984i	$-0.819564\pi$
$13$	−79.0820	−1.68719	−0.843593	+	0.536984i	$0.819564\pi$
$14$	0	0
$15$	32.2918	0.555847
$16$	0	0
$17$	−77.2361	−1.10191	−0.550956	−	0.834534i	$-0.685737\pi$
$17$	−77.2361	−1.10191	−0.550956	+	0.834534i	$0.685737\pi$
$18$	0	0
$19$	−50.7902	−0.613267	−0.306634	−	0.951828i	$-0.599203\pi$
$19$	−50.7902	−0.613267	−0.306634	+	0.951828i	$0.599203\pi$
$20$	0	0
$21$	31.9574	0.332080
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−9.13777	−0.0731021
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	12.7477	0.0816270	0.0408135	−	0.999167i	$-0.487005\pi$
$29$	12.7477	0.0816270	0.0408135	+	0.999167i	$0.487005\pi$
$30$	0	0
$31$	12.4133	0.0719192	0.0359596	−	0.999353i	$-0.488551\pi$
$31$	12.4133	0.0719192	0.0359596	+	0.999353i	$0.488551\pi$
$32$	0	0
$33$	−157.082	−0.828620
$34$	0	0
$35$	114.663	0.553757
$36$	0	0
$37$	−73.1084	−0.324836	−0.162418	−	0.986722i	$-0.551929\pi$
$37$	−73.1084	−0.324836	−0.162418	+	0.986722i	$0.551929\pi$
$38$	0	0
$39$	237.246	0.974097
$40$	0	0
$41$	−38.8916	−0.148143	−0.0740714	−	0.997253i	$-0.523599\pi$
$41$	−38.8916	−0.148143	−0.0740714	+	0.997253i	$0.523599\pi$
$42$	0	0
$43$	−171.787	−0.609239	−0.304620	−	0.952474i	$-0.598529\pi$
$43$	−171.787	−0.609239	−0.304620	+	0.952474i	$0.598529\pi$
$44$	0	0
$45$	−96.8754	−0.320918
$46$	0	0
$47$	−614.545	−1.90725	−0.953623	−	0.301003i	$-0.902679\pi$
$47$	−614.545	−1.90725	−0.953623	+	0.301003i	$0.902679\pi$
$48$	0	0
$49$	−229.525	−0.669168
$50$	0	0
$51$	231.708	0.636189
$52$	0	0
$53$	269.597	0.698716	0.349358	−	0.936989i	$-0.386400\pi$
$53$	269.597	0.698716	0.349358	+	0.936989i	$0.386400\pi$
$54$	0	0
$55$	−563.607	−1.38176
$56$	0	0
$57$	152.371	0.354070
$58$	0	0
$59$	534.768	1.18001	0.590007	−	0.807398i	$-0.299125\pi$
$59$	534.768	1.18001	0.590007	+	0.807398i	$0.299125\pi$
$60$	0	0
$61$	−838.604	−1.76020	−0.880100	−	0.474788i	$-0.842525\pi$
$61$	−838.604	−1.76020	−0.880100	+	0.474788i	$0.842525\pi$
$62$	0	0
$63$	−95.8723	−0.191727
$64$	0	0
$65$	851.234	1.62435
$66$	0	0
$67$	448.180	0.817223	0.408612	−	0.912708i	$-0.366013\pi$
$67$	448.180	0.817223	0.408612	+	0.912708i	$0.366013\pi$
$68$	0	0
$69$	−69.0000	−0.120386
$70$	0	0
$71$	−628.604	−1.05073	−0.525363	−	0.850878i	$-0.676070\pi$
$71$	−628.604	−1.05073	−0.525363	+	0.850878i	$0.676070\pi$
$72$	0	0
$73$	925.266	1.48348	0.741741	−	0.670686i	$-0.234000\pi$
$73$	925.266	1.48348	0.741741	+	0.670686i	$0.234000\pi$
$74$	0	0
$75$	27.4133	0.0422055
$76$	0	0
$77$	−557.771	−0.825505
$78$	0	0
$79$	963.479	1.37215	0.686075	−	0.727531i	$-0.259332\pi$
$79$	963.479	1.37215	0.686075	+	0.727531i	$0.259332\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−133.358	−0.176360	−0.0881801	−	0.996105i	$-0.528105\pi$
$83$	−133.358	−0.176360	−0.0881801	+	0.996105i	$0.528105\pi$
$84$	0	0
$85$	831.364	1.06087
$86$	0	0
$87$	−38.2430	−0.0471274
$88$	0	0
$89$	−778.581	−0.927297	−0.463649	−	0.886019i	$-0.653460\pi$
$89$	−778.581	−0.927297	−0.463649	+	0.886019i	$0.653460\pi$
$90$	0	0
$91$	842.420	0.970435
$92$	0	0
$93$	−37.2399	−0.0415226
$94$	0	0
$95$	546.703	0.590426
$96$	0	0
$97$	1603.57	1.67853	0.839266	−	0.543721i	$-0.182985\pi$
$97$	1603.57	1.67853	0.839266	+	0.543721i	$0.182985\pi$
$98$	0	0
$99$	471.246	0.478404
$100$	0	0
$101$	1229.04	1.21084	0.605418	−	0.795908i	$-0.293006\pi$
$101$	1229.04	1.21084	0.605418	+	0.795908i	$0.293006\pi$
$102$	0	0
$103$	−215.066	−0.205738	−0.102869	−	0.994695i	$-0.532802\pi$
$103$	−215.066	−0.205738	−0.102869	+	0.994695i	$0.532802\pi$
$104$	0	0
$105$	−343.988	−0.319712
$106$	0	0
$107$	1586.58	1.43346	0.716730	−	0.697351i	$-0.245638\pi$
$107$	1586.58	1.43346	0.716730	+	0.697351i	$0.245638\pi$
$108$	0	0
$109$	822.255	0.722549	0.361274	−	0.932460i	$-0.382342\pi$
$109$	822.255	0.722549	0.361274	+	0.932460i	$0.382342\pi$
$110$	0	0
$111$	219.325	0.187544
$112$	0	0
$113$	19.7005	0.0164006	0.00820031	−	0.999966i	$-0.497390\pi$
$113$	19.7005	0.0164006	0.00820031	+	0.999966i	$0.497390\pi$
$114$	0	0
$115$	−247.570	−0.200748
$116$	0	0
$117$	−711.738	−0.562395
$118$	0	0
$119$	822.755	0.633797
$120$	0	0
$121$	1410.64	1.05984
$122$	0	0
$123$	116.675	0.0855303
$124$	0	0
$125$	1443.85	1.03313
$126$	0	0
$127$	−1143.23	−0.798784	−0.399392	−	0.916780i	$-0.630779\pi$
$127$	−1143.23	−0.798784	−0.399392	+	0.916780i	$0.630779\pi$
$128$	0	0
$129$	515.361	0.351745
$130$	0	0
$131$	−1154.64	−0.770085	−0.385043	−	0.922899i	$-0.625813\pi$
$131$	−1154.64	−0.770085	−0.385043	+	0.922899i	$0.625813\pi$
$132$	0	0
$133$	541.042	0.352739
$134$	0	0
$135$	290.626	0.185282
$136$	0	0
$137$	−2153.10	−1.34271	−0.671356	−	0.741135i	$-0.734288\pi$
$137$	−2153.10	−1.34271	−0.671356	+	0.741135i	$0.734288\pi$
$138$	0	0
$139$	2392.16	1.45971	0.729857	−	0.683600i	$-0.239587\pi$
$139$	2392.16	1.45971	0.729857	+	0.683600i	$0.239587\pi$
$140$	0	0
$141$	1843.63	1.10115
$142$	0	0
$143$	−4140.79	−2.42147
$144$	0	0
$145$	−137.215	−0.0785868
$146$	0	0
$147$	688.574	0.386345
$148$	0	0
$149$	−2259.80	−1.24248	−0.621242	−	0.783619i	$-0.713372\pi$
$149$	−2259.80	−1.24248	−0.621242	+	0.783619i	$0.713372\pi$
$150$	0	0
$151$	2410.84	1.29928	0.649639	−	0.760243i	$-0.274920\pi$
$151$	2410.84	1.29928	0.649639	+	0.760243i	$0.274920\pi$
$152$	0	0
$153$	−695.125	−0.367304
$154$	0	0
$155$	−133.616	−0.0692406
$156$	0	0
$157$	−614.960	−0.312606	−0.156303	−	0.987709i	$-0.549958\pi$
$157$	−614.960	−0.312606	−0.156303	+	0.987709i	$0.549958\pi$
$158$	0	0
$159$	−808.790	−0.403404
$160$	0	0
$161$	−245.007	−0.119933
$162$	0	0
$163$	50.8127	0.0244169	0.0122085	−	0.999925i	$-0.496114\pi$
$163$	50.8127	0.0244169	0.0122085	+	0.999925i	$0.496114\pi$
$164$	0	0
$165$	1690.82	0.797759
$166$	0	0
$167$	2000.84	0.927125	0.463562	−	0.886064i	$-0.346571\pi$
$167$	2000.84	0.927125	0.463562	+	0.886064i	$0.346571\pi$
$168$	0	0
$169$	4056.97	1.84659
$170$	0	0
$171$	−457.112	−0.204422
$172$	0	0
$173$	−3796.78	−1.66858	−0.834288	−	0.551330i	$-0.814121\pi$
$173$	−3796.78	−1.66858	−0.834288	+	0.551330i	$0.814121\pi$
$174$	0	0
$175$	97.3398	0.0420469
$176$	0	0
$177$	−1604.30	−0.681281
$178$	0	0
$179$	3084.16	1.28783	0.643913	−	0.765099i	$-0.277310\pi$
$179$	3084.16	1.28783	0.643913	+	0.765099i	$0.277310\pi$
$180$	0	0
$181$	4733.12	1.94370	0.971851	−	0.235597i	$-0.0757047\pi$
$181$	4733.12	1.94370	0.971851	+	0.235597i	$0.0757047\pi$
$182$	0	0
$183$	2515.81	1.01625
$184$	0	0
$185$	786.933	0.312738
$186$	0	0
$187$	−4044.13	−1.58148
$188$	0	0
$189$	287.617	0.110693
$190$	0	0
$191$	1293.13	0.489885	0.244942	−	0.969538i	$-0.421231\pi$
$191$	1293.13	0.489885	0.244942	+	0.969538i	$0.421231\pi$
$192$	0	0
$193$	−1171.75	−0.437019	−0.218510	−	0.975835i	$-0.570119\pi$
$193$	−1171.75	−0.437019	−0.218510	+	0.975835i	$0.570119\pi$
$194$	0	0
$195$	−2553.70	−0.937817
$196$	0	0
$197$	4559.19	1.64888	0.824438	−	0.565952i	$-0.191491\pi$
$197$	4559.19	1.64888	0.824438	+	0.565952i	$0.191491\pi$
$198$	0	0
$199$	473.721	0.168750	0.0843748	−	0.996434i	$-0.473111\pi$
$199$	473.721	0.168750	0.0843748	+	0.996434i	$0.473111\pi$
$200$	0	0
$201$	−1344.54	−0.471824
$202$	0	0
$203$	−135.794	−0.0469502
$204$	0	0
$205$	418.627	0.142625
$206$	0	0
$207$	207.000	0.0695048
$208$	0	0
$209$	−2659.41	−0.880169
$210$	0	0
$211$	1423.56	0.464465	0.232232	−	0.972660i	$-0.425397\pi$
$211$	1423.56	0.464465	0.232232	+	0.972660i	$0.425397\pi$
$212$	0	0
$213$	1885.81	0.606637
$214$	0	0
$215$	1849.11	0.586548
$216$	0	0
$217$	−132.232	−0.0413665
$218$	0	0
$219$	−2775.80	−0.856489
$220$	0	0
$221$	6107.99	1.85913
$222$	0	0
$223$	−6088.63	−1.82836	−0.914181	−	0.405305i	$-0.867165\pi$
$223$	−6088.63	−1.82836	−0.914181	+	0.405305i	$0.867165\pi$
$224$	0	0
$225$	−82.2399	−0.0243674
$226$	0	0
$227$	4463.66	1.30513	0.652563	−	0.757734i	$-0.273694\pi$
$227$	4463.66	1.30513	0.652563	+	0.757734i	$0.273694\pi$
$228$	0	0
$229$	1298.87	0.374812	0.187406	−	0.982283i	$-0.439992\pi$
$229$	1298.87	0.374812	0.187406	+	0.982283i	$0.439992\pi$
$230$	0	0
$231$	1673.31	0.476606
$232$	0	0
$233$	−1257.41	−0.353544	−0.176772	−	0.984252i	$-0.556566\pi$
$233$	−1257.41	−0.353544	−0.176772	+	0.984252i	$0.556566\pi$
$234$	0	0
$235$	6614.92	1.83621
$236$	0	0
$237$	−2890.44	−0.792211
$238$	0	0
$239$	−3473.96	−0.940217	−0.470108	−	0.882609i	$-0.655785\pi$
$239$	−3473.96	−0.940217	−0.470108	+	0.882609i	$0.655785\pi$
$240$	0	0
$241$	−2140.92	−0.572235	−0.286118	−	0.958195i	$-0.592365\pi$
$241$	−2140.92	−0.572235	−0.286118	+	0.958195i	$0.592365\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	2470.59	0.644245
$246$	0	0
$247$	4016.60	1.03470
$248$	0	0
$249$	400.073	0.101822
$250$	0	0
$251$	5613.17	1.41155	0.705777	−	0.708434i	$-0.250598\pi$
$251$	5613.17	1.41155	0.705777	+	0.708434i	$0.250598\pi$
$252$	0	0
$253$	1204.30	0.299263
$254$	0	0
$255$	−2494.09	−0.612494
$256$	0	0
$257$	1017.98	0.247082	0.123541	−	0.992339i	$-0.460575\pi$
$257$	1017.98	0.247082	0.123541	+	0.992339i	$0.460575\pi$
$258$	0	0
$259$	778.785	0.186839
$260$	0	0
$261$	114.729	0.0272090
$262$	0	0
$263$	995.531	0.233411	0.116705	−	0.993167i	$-0.462767\pi$
$263$	995.531	0.233411	0.116705	+	0.993167i	$0.462767\pi$
$264$	0	0
$265$	−2901.92	−0.672693
$266$	0	0
$267$	2335.74	0.535375
$268$	0	0
$269$	−3568.95	−0.808932	−0.404466	−	0.914553i	$-0.632543\pi$
$269$	−3568.95	−0.808932	−0.404466	+	0.914553i	$0.632543\pi$
$270$	0	0
$271$	6294.26	1.41088	0.705441	−	0.708768i	$-0.250749\pi$
$271$	6294.26	1.41088	0.705441	+	0.708768i	$0.250749\pi$
$272$	0	0
$273$	−2527.26	−0.560281
$274$	0	0
$275$	−478.460	−0.104917
$276$	0	0
$277$	−2263.65	−0.491009	−0.245505	−	0.969395i	$-0.578954\pi$
$277$	−2263.65	−0.491009	−0.245505	+	0.969395i	$0.578954\pi$
$278$	0	0
$279$	111.720	0.0239731
$280$	0	0
$281$	−505.900	−0.107400	−0.0537001	−	0.998557i	$-0.517102\pi$
$281$	−505.900	−0.107400	−0.0537001	+	0.998557i	$0.517102\pi$
$282$	0	0
$283$	7018.20	1.47417	0.737083	−	0.675802i	$-0.236203\pi$
$283$	7018.20	1.47417	0.737083	+	0.675802i	$0.236203\pi$
$284$	0	0
$285$	−1640.11	−0.340883
$286$	0	0
$287$	414.292	0.0852087
$288$	0	0
$289$	1052.41	0.214209
$290$	0	0
$291$	−4810.70	−0.969101
$292$	0	0
$293$	−9488.82	−1.89196	−0.945978	−	0.324232i	$-0.894894\pi$
$293$	−9488.82	−1.89196	−0.945978	+	0.324232i	$0.894894\pi$
$294$	0	0
$295$	−5756.20	−1.13606
$296$	0	0
$297$	−1413.74	−0.276207
$298$	0	0
$299$	−1818.89	−0.351802
$300$	0	0
$301$	1829.96	0.350422
$302$	0	0
$303$	−3687.13	−0.699076
$304$	0	0
$305$	9026.67	1.69464
$306$	0	0
$307$	−4660.58	−0.866427	−0.433214	−	0.901291i	$-0.642620\pi$
$307$	−4660.58	−0.866427	−0.433214	+	0.901291i	$0.642620\pi$
$308$	0	0
$309$	645.197	0.118783
$310$	0	0
$311$	−4378.51	−0.798336	−0.399168	−	0.916878i	$-0.630701\pi$
$311$	−4378.51	−0.798336	−0.399168	+	0.916878i	$0.630701\pi$
$312$	0	0
$313$	−4444.17	−0.802553	−0.401277	−	0.915957i	$-0.631433\pi$
$313$	−4444.17	−0.802553	−0.401277	+	0.915957i	$0.631433\pi$
$314$	0	0
$315$	1031.96	0.184586
$316$	0	0
$317$	7881.61	1.39645	0.698226	−	0.715878i	$-0.253973\pi$
$317$	7881.61	1.39645	0.698226	+	0.715878i	$0.253973\pi$
$318$	0	0
$319$	667.477	0.117152
$320$	0	0
$321$	−4759.73	−0.827609
$322$	0	0
$323$	3922.84	0.675767
$324$	0	0
$325$	722.633	0.123337
$326$	0	0
$327$	−2466.77	−0.417164
$328$	0	0
$329$	6546.42	1.09701
$330$	0	0
$331$	127.060	0.0210993	0.0105497	−	0.999944i	$-0.496642\pi$
$331$	127.060	0.0210993	0.0105497	+	0.999944i	$0.496642\pi$
$332$	0	0
$333$	−657.975	−0.108279
$334$	0	0
$335$	−4824.18	−0.786786
$336$	0	0
$337$	−1739.07	−0.281107	−0.140554	−	0.990073i	$-0.544888\pi$
$337$	−1739.07	−0.281107	−0.140554	+	0.990073i	$0.544888\pi$
$338$	0	0
$339$	−59.1016	−0.00946890
$340$	0	0
$341$	649.969	0.103219
$342$	0	0
$343$	6098.81	0.960072
$344$	0	0
$345$	742.711	0.115902
$346$	0	0
$347$	−7551.30	−1.16823	−0.584114	−	0.811672i	$-0.698558\pi$
$347$	−7551.30	−1.16823	−0.584114	+	0.811672i	$0.698558\pi$
$348$	0	0
$349$	−6237.28	−0.956659	−0.478329	−	0.878180i	$-0.658758\pi$
$349$	−6237.28	−0.956659	−0.478329	+	0.878180i	$0.658758\pi$
$350$	0	0
$351$	2135.22	0.324699
$352$	0	0
$353$	1788.57	0.269678	0.134839	−	0.990868i	$-0.456948\pi$
$353$	1788.57	0.269678	0.134839	+	0.990868i	$0.456948\pi$
$354$	0	0
$355$	6766.25	1.01159
$356$	0	0
$357$	−2468.27	−0.365923
$358$	0	0
$359$	−2234.70	−0.328531	−0.164266	−	0.986416i	$-0.552525\pi$
$359$	−2234.70	−0.328531	−0.164266	+	0.986416i	$0.552525\pi$
$360$	0	0
$361$	−4279.35	−0.623903
$362$	0	0
$363$	−4231.92	−0.611896
$364$	0	0
$365$	−9959.50	−1.42823
$366$	0	0
$367$	−10551.7	−1.50080	−0.750399	−	0.660985i	$-0.770139\pi$
$367$	−10551.7	−1.50080	−0.750399	+	0.660985i	$0.770139\pi$
$368$	0	0
$369$	−350.025	−0.0493809
$370$	0	0
$371$	−2871.87	−0.401887
$372$	0	0
$373$	−996.068	−0.138269	−0.0691347	−	0.997607i	$-0.522024\pi$
$373$	−996.068	−0.138269	−0.0691347	+	0.997607i	$0.522024\pi$
$374$	0	0
$375$	−4331.55	−0.596481
$376$	0	0
$377$	−1008.11	−0.137720
$378$	0	0
$379$	11117.0	1.50670	0.753351	−	0.657619i	$-0.228436\pi$
$379$	11117.0	1.50670	0.753351	+	0.657619i	$0.228436\pi$
$380$	0	0
$381$	3429.70	0.461178
$382$	0	0
$383$	−5315.09	−0.709108	−0.354554	−	0.935035i	$-0.615367\pi$
$383$	−5315.09	−0.709108	−0.354554	+	0.935035i	$0.615367\pi$
$384$	0	0
$385$	6003.81	0.794759
$386$	0	0
$387$	−1546.08	−0.203080
$388$	0	0
$389$	−8438.06	−1.09981	−0.549906	−	0.835227i	$-0.685336\pi$
$389$	−8438.06	−1.09981	−0.549906	+	0.835227i	$0.685336\pi$
$390$	0	0
$391$	−1776.43	−0.229764
$392$	0	0
$393$	3463.91	0.444609
$394$	0	0
$395$	−10370.8	−1.32104
$396$	0	0
$397$	11582.0	1.46419	0.732094	−	0.681204i	$-0.238543\pi$
$397$	11582.0	1.46419	0.732094	+	0.681204i	$0.238543\pi$
$398$	0	0
$399$	−1623.13	−0.203654
$400$	0	0
$401$	4267.85	0.531487	0.265744	−	0.964044i	$-0.414383\pi$
$401$	4267.85	0.531487	0.265744	+	0.964044i	$0.414383\pi$
$402$	0	0
$403$	−981.669	−0.121341
$404$	0	0
$405$	−871.878	−0.106973
$406$	0	0
$407$	−3828.00	−0.466209
$408$	0	0
$409$	6902.51	0.834491	0.417246	−	0.908794i	$-0.362995\pi$
$409$	6902.51	0.834491	0.417246	+	0.908794i	$0.362995\pi$
$410$	0	0
$411$	6459.29	0.775215
$412$	0	0
$413$	−5696.60	−0.678720
$414$	0	0
$415$	1435.45	0.169792
$416$	0	0
$417$	−7176.47	−0.842766
$418$	0	0
$419$	2241.24	0.261316	0.130658	−	0.991427i	$-0.458291\pi$
$419$	2241.24	0.261316	0.130658	+	0.991427i	$0.458291\pi$
$420$	0	0
$421$	9117.52	1.05549	0.527744	−	0.849403i	$-0.323038\pi$
$421$	9117.52	1.05549	0.527744	+	0.849403i	$0.323038\pi$
$422$	0	0
$423$	−5530.90	−0.635749
$424$	0	0
$425$	705.765	0.0805521
$426$	0	0
$427$	8933.21	1.01243
$428$	0	0
$429$	12422.4	1.39804
$430$	0	0
$431$	1366.55	0.152725	0.0763623	−	0.997080i	$-0.475669\pi$
$431$	1366.55	0.152725	0.0763623	+	0.997080i	$0.475669\pi$
$432$	0	0
$433$	3511.21	0.389695	0.194848	−	0.980834i	$-0.437579\pi$
$433$	3511.21	0.389695	0.194848	+	0.980834i	$0.437579\pi$
$434$	0	0
$435$	411.645	0.0453721
$436$	0	0
$437$	−1168.18	−0.127875
$438$	0	0
$439$	−7032.17	−0.764526	−0.382263	−	0.924054i	$-0.624855\pi$
$439$	−7032.17	−0.764526	−0.382263	+	0.924054i	$0.624855\pi$
$440$	0	0
$441$	−2065.72	−0.223056
$442$	0	0
$443$	−8748.68	−0.938289	−0.469145	−	0.883121i	$-0.655438\pi$
$443$	−8748.68	−0.938289	−0.469145	+	0.883121i	$0.655438\pi$
$444$	0	0
$445$	8380.60	0.892760
$446$	0	0
$447$	6779.40	0.717348
$448$	0	0
$449$	−805.208	−0.0846327	−0.0423164	−	0.999104i	$-0.513474\pi$
$449$	−805.208	−0.0846327	−0.0423164	+	0.999104i	$0.513474\pi$
$450$	0	0
$451$	−2036.39	−0.212616
$452$	0	0
$453$	−7232.51	−0.750139
$454$	0	0
$455$	−9067.75	−0.934291
$456$	0	0
$457$	−10496.9	−1.07445	−0.537226	−	0.843438i	$-0.680528\pi$
$457$	−10496.9	−1.07445	−0.537226	+	0.843438i	$0.680528\pi$
$458$	0	0
$459$	2085.37	0.212063
$460$	0	0
$461$	−3325.68	−0.335992	−0.167996	−	0.985788i	$-0.553730\pi$
$461$	−3325.68	−0.335992	−0.167996	+	0.985788i	$0.553730\pi$
$462$	0	0
$463$	840.876	0.0844035	0.0422018	−	0.999109i	$-0.486563\pi$
$463$	840.876	0.0844035	0.0422018	+	0.999109i	$0.486563\pi$
$464$	0	0
$465$	400.848	0.0399761
$466$	0	0
$467$	−8450.82	−0.837382	−0.418691	−	0.908129i	$-0.637511\pi$
$467$	−8450.82	−0.837382	−0.418691	+	0.908129i	$0.637511\pi$
$468$	0	0
$469$	−4774.23	−0.470050
$470$	0	0
$471$	1844.88	0.180483
$472$	0	0
$473$	−8994.89	−0.874388
$474$	0	0
$475$	464.109	0.0448312
$476$	0	0
$477$	2426.37	0.232905
$478$	0	0
$479$	5686.86	0.542462	0.271231	−	0.962514i	$-0.412569\pi$
$479$	5686.86	0.542462	0.271231	+	0.962514i	$0.412569\pi$
$480$	0	0
$481$	5781.56	0.548059
$482$	0	0
$483$	735.021	0.0692435
$484$	0	0
$485$	−17260.7	−1.61602
$486$	0	0
$487$	13768.2	1.28110	0.640551	−	0.767916i	$-0.278706\pi$
$487$	13768.2	1.28110	0.640551	+	0.767916i	$0.278706\pi$
$488$	0	0
$489$	−152.438	−0.0140971
$490$	0	0
$491$	−8480.14	−0.779436	−0.389718	−	0.920934i	$-0.627428\pi$
$491$	−8480.14	−0.779436	−0.389718	+	0.920934i	$0.627428\pi$
$492$	0	0
$493$	−984.580	−0.0899457
$494$	0	0
$495$	−5072.46	−0.460586
$496$	0	0
$497$	6696.19	0.604356
$498$	0	0
$499$	−12472.5	−1.11893	−0.559466	−	0.828853i	$-0.688994\pi$
$499$	−12472.5	−1.11893	−0.559466	+	0.828853i	$0.688994\pi$
$500$	0	0
$501$	−6002.53	−0.535276
$502$	0	0
$503$	4676.36	0.414530	0.207265	−	0.978285i	$-0.433544\pi$
$503$	4676.36	0.414530	0.207265	+	0.978285i	$0.433544\pi$
$504$	0	0
$505$	−13229.3	−1.16574
$506$	0	0
$507$	−12170.9	−1.06613
$508$	0	0
$509$	5925.58	0.516005	0.258003	−	0.966144i	$-0.416936\pi$
$509$	5925.58	0.516005	0.258003	+	0.966144i	$0.416936\pi$
$510$	0	0
$511$	−9856.38	−0.853269
$512$	0	0
$513$	1371.34	0.118023
$514$	0	0
$515$	2314.95	0.198076
$516$	0	0
$517$	−32178.0	−2.73730
$518$	0	0
$519$	11390.3	0.963352
$520$	0	0
$521$	2626.25	0.220841	0.110421	−	0.993885i	$-0.464780\pi$
$521$	2626.25	0.220841	0.110421	+	0.993885i	$0.464780\pi$
$522$	0	0
$523$	−372.504	−0.0311443	−0.0155721	−	0.999879i	$-0.504957\pi$
$523$	−372.504	−0.0311443	−0.0155721	+	0.999879i	$0.504957\pi$
$524$	0	0
$525$	−292.020	−0.0242758
$526$	0	0
$527$	−958.755	−0.0792486
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	4812.91	0.393338
$532$	0	0
$533$	3075.63	0.249944
$534$	0	0
$535$	−17077.8	−1.38007
$536$	0	0
$537$	−9252.47	−0.743526
$538$	0	0
$539$	−12018.1	−0.960399
$540$	0	0
$541$	12899.2	1.02510	0.512550	−	0.858657i	$-0.328701\pi$
$541$	12899.2	1.02510	0.512550	+	0.858657i	$0.328701\pi$
$542$	0	0
$543$	−14199.4	−1.12220
$544$	0	0
$545$	−8850.70	−0.695637
$546$	0	0
$547$	22144.7	1.73097	0.865485	−	0.500936i	$-0.167011\pi$
$547$	22144.7	1.73097	0.865485	+	0.500936i	$0.167011\pi$
$548$	0	0
$549$	−7547.43	−0.586733
$550$	0	0
$551$	−647.457	−0.0500592
$552$	0	0
$553$	−10263.4	−0.789233
$554$	0	0
$555$	−2360.80	−0.180559
$556$	0	0
$557$	−10236.7	−0.778716	−0.389358	−	0.921086i	$-0.627303\pi$
$557$	−10236.7	−0.778716	−0.389358	+	0.921086i	$0.627303\pi$
$558$	0	0
$559$	13585.3	1.02790
$560$	0	0
$561$	12132.4	0.913066
$562$	0	0
$563$	−5750.83	−0.430495	−0.215247	−	0.976560i	$-0.569056\pi$
$563$	−5750.83	−0.430495	−0.215247	+	0.976560i	$0.569056\pi$
$564$	0	0
$565$	−212.055	−0.0157898
$566$	0	0
$567$	−862.851	−0.0639088
$568$	0	0
$569$	−8285.10	−0.610421	−0.305210	−	0.952285i	$-0.598727\pi$
$569$	−8285.10	−0.610421	−0.305210	+	0.952285i	$0.598727\pi$
$570$	0	0
$571$	19799.8	1.45113	0.725565	−	0.688154i	$-0.241579\pi$
$571$	19799.8	1.45113	0.725565	+	0.688154i	$0.241579\pi$
$572$	0	0
$573$	−3879.40	−0.282835
$574$	0	0
$575$	−210.169	−0.0152428
$576$	0	0
$577$	−250.868	−0.0181001	−0.00905006	−	0.999959i	$-0.502881\pi$
$577$	−250.868	−0.0181001	−0.00905006	+	0.999959i	$0.502881\pi$
$578$	0	0
$579$	3515.26	0.252313
$580$	0	0
$581$	1420.59	0.101439
$582$	0	0
$583$	14116.3	1.00281
$584$	0	0
$585$	7661.10	0.541449
$586$	0	0
$587$	9767.11	0.686766	0.343383	−	0.939195i	$-0.388427\pi$
$587$	9767.11	0.686766	0.343383	+	0.939195i	$0.388427\pi$
$588$	0	0
$589$	−630.475	−0.0441057
$590$	0	0
$591$	−13677.6	−0.951979
$592$	0	0
$593$	−18853.8	−1.30562	−0.652810	−	0.757521i	$-0.726410\pi$
$593$	−18853.8	−1.30562	−0.652810	+	0.757521i	$0.726410\pi$
$594$	0	0
$595$	−8856.08	−0.610192
$596$	0	0
$597$	−1421.16	−0.0974276
$598$	0	0
$599$	11356.0	0.774615	0.387307	−	0.921951i	$-0.373405\pi$
$599$	11356.0	0.774615	0.387307	+	0.921951i	$0.373405\pi$
$600$	0	0
$601$	20359.1	1.38180	0.690902	−	0.722948i	$-0.257213\pi$
$601$	20359.1	1.38180	0.690902	+	0.722948i	$0.257213\pi$
$602$	0	0
$603$	4033.62	0.272408
$604$	0	0
$605$	−15184.0	−1.02036
$606$	0	0
$607$	−5975.14	−0.399545	−0.199772	−	0.979842i	$-0.564020\pi$
$607$	−5975.14	−0.399545	−0.199772	+	0.979842i	$0.564020\pi$
$608$	0	0
$609$	407.383	0.0271067
$610$	0	0
$611$	48599.5	3.21788
$612$	0	0
$613$	16997.4	1.11993	0.559967	−	0.828515i	$-0.310814\pi$
$613$	16997.4	1.11993	0.559967	+	0.828515i	$0.310814\pi$
$614$	0	0
$615$	−1255.88	−0.0823447
$616$	0	0
$617$	−13096.0	−0.854497	−0.427249	−	0.904134i	$-0.640517\pi$
$617$	−13096.0	−0.854497	−0.427249	+	0.904134i	$0.640517\pi$
$618$	0	0
$619$	−702.058	−0.0455866	−0.0227933	−	0.999740i	$-0.507256\pi$
$619$	−702.058	−0.0455866	−0.0227933	+	0.999740i	$0.507256\pi$
$620$	0	0
$621$	−621.000	−0.0401286
$622$	0	0
$623$	8293.82	0.533362
$624$	0	0
$625$	−14399.3	−0.921554
$626$	0	0
$627$	7978.23	0.508166
$628$	0	0
$629$	5646.60	0.357941
$630$	0	0
$631$	−23871.5	−1.50604	−0.753019	−	0.657999i	$-0.771403\pi$
$631$	−23871.5	−1.50604	−0.753019	+	0.657999i	$0.771403\pi$
$632$	0	0
$633$	−4270.69	−0.268159
$634$	0	0
$635$	12305.7	0.769034
$636$	0	0
$637$	18151.3	1.12901
$638$	0	0
$639$	−5657.43	−0.350242
$640$	0	0
$641$	24172.6	1.48949	0.744743	−	0.667351i	$-0.232572\pi$
$641$	24172.6	1.48949	0.744743	+	0.667351i	$0.232572\pi$
$642$	0	0
$643$	−13648.6	−0.837090	−0.418545	−	0.908196i	$-0.637460\pi$
$643$	−13648.6	−0.837090	−0.418545	+	0.908196i	$0.637460\pi$
$644$	0	0
$645$	−5547.32	−0.338644
$646$	0	0
$647$	7009.18	0.425903	0.212952	−	0.977063i	$-0.431692\pi$
$647$	7009.18	0.425903	0.212952	+	0.977063i	$0.431692\pi$
$648$	0	0
$649$	28000.8	1.69357
$650$	0	0
$651$	396.697	0.0238829
$652$	0	0
$653$	4533.57	0.271688	0.135844	−	0.990730i	$-0.456625\pi$
$653$	4533.57	0.271688	0.135844	+	0.990730i	$0.456625\pi$
$654$	0	0
$655$	12428.4	0.741404
$656$	0	0
$657$	8327.40	0.494494
$658$	0	0
$659$	7112.62	0.420438	0.210219	−	0.977654i	$-0.432582\pi$
$659$	7112.62	0.420438	0.210219	+	0.977654i	$0.432582\pi$
$660$	0	0
$661$	16088.6	0.946709	0.473354	−	0.880872i	$-0.343043\pi$
$661$	16088.6	0.946709	0.473354	+	0.880872i	$0.343043\pi$
$662$	0	0
$663$	−18324.0	−1.07337
$664$	0	0
$665$	−5823.74	−0.339601
$666$	0	0
$667$	293.196	0.0170204
$668$	0	0
$669$	18265.9	1.05561
$670$	0	0
$671$	−43909.9	−2.52626
$672$	0	0
$673$	22629.7	1.29615	0.648076	−	0.761576i	$-0.275574\pi$
$673$	22629.7	1.29615	0.648076	+	0.761576i	$0.275574\pi$
$674$	0	0
$675$	246.720	0.0140685
$676$	0	0
$677$	−15444.0	−0.876749	−0.438375	−	0.898792i	$-0.644446\pi$
$677$	−15444.0	−0.876749	−0.438375	+	0.898792i	$0.644446\pi$
$678$	0	0
$679$	−17082.0	−0.965458
$680$	0	0
$681$	−13391.0	−0.753515
$682$	0	0
$683$	−834.894	−0.0467736	−0.0233868	−	0.999726i	$-0.507445\pi$
$683$	−834.894	−0.0467736	−0.0233868	+	0.999726i	$0.507445\pi$
$684$	0	0
$685$	23175.8	1.29270
$686$	0	0
$687$	−3896.62	−0.216398
$688$	0	0
$689$	−21320.3	−1.17886
$690$	0	0
$691$	−4982.99	−0.274330	−0.137165	−	0.990548i	$-0.543799\pi$
$691$	−4982.99	−0.274330	−0.137165	+	0.990548i	$0.543799\pi$
$692$	0	0
$693$	−5019.94	−0.275168
$694$	0	0
$695$	−25749.0	−1.40535
$696$	0	0
$697$	3003.84	0.163240
$698$	0	0
$699$	3772.23	0.204119
$700$	0	0
$701$	21916.9	1.18087	0.590436	−	0.807084i	$-0.298956\pi$
$701$	21916.9	1.18087	0.590436	+	0.807084i	$0.298956\pi$
$702$	0	0
$703$	3713.19	0.199211
$704$	0	0
$705$	−19844.8	−1.06014
$706$	0	0
$707$	−13092.4	−0.696448
$708$	0	0
$709$	−30858.3	−1.63456	−0.817282	−	0.576237i	$-0.804520\pi$
$709$	−30858.3	−1.63456	−0.817282	+	0.576237i	$0.804520\pi$
$710$	0	0
$711$	8671.31	0.457383
$712$	0	0
$713$	285.506	0.0149962
$714$	0	0
$715$	44571.2	2.33128
$716$	0	0
$717$	10421.9	0.542834
$718$	0	0
$719$	−19500.3	−1.01146	−0.505728	−	0.862693i	$-0.668776\pi$
$719$	−19500.3	−1.01146	−0.505728	+	0.862693i	$0.668776\pi$
$720$	0	0
$721$	2290.98	0.118337
$722$	0	0
$723$	6422.75	0.330380
$724$	0	0
$725$	−116.485	−0.00596711
$726$	0	0
$727$	−26922.8	−1.37347	−0.686735	−	0.726908i	$-0.740957\pi$
$727$	−26922.8	−1.37347	−0.686735	+	0.726908i	$0.740957\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	13268.2	0.671328
$732$	0	0
$733$	−22120.5	−1.11465	−0.557326	−	0.830294i	$-0.688173\pi$
$733$	−22120.5	−1.11465	−0.557326	+	0.830294i	$0.688173\pi$
$734$	0	0
$735$	−7411.77	−0.371955
$736$	0	0
$737$	23467.0	1.17289
$738$	0	0
$739$	27225.8	1.35523	0.677617	−	0.735415i	$-0.263013\pi$
$739$	27225.8	1.35523	0.677617	+	0.735415i	$0.263013\pi$
$740$	0	0
$741$	−12049.8	−0.597382
$742$	0	0
$743$	191.429	0.00945201	0.00472600	−	0.999989i	$-0.498496\pi$
$743$	191.429	0.00945201	0.00472600	+	0.999989i	$0.498496\pi$
$744$	0	0
$745$	24324.3	1.19621
$746$	0	0
$747$	−1200.22	−0.0587867
$748$	0	0
$749$	−16901.0	−0.824497
$750$	0	0
$751$	876.709	0.0425986	0.0212993	−	0.999773i	$-0.493220\pi$
$751$	876.709	0.0425986	0.0212993	+	0.999773i	$0.493220\pi$
$752$	0	0
$753$	−16839.5	−0.814961
$754$	0	0
$755$	−25950.1	−1.25089
$756$	0	0
$757$	−26681.0	−1.28103	−0.640513	−	0.767948i	$-0.721278\pi$
$757$	−26681.0	−1.28103	−0.640513	+	0.767948i	$0.721278\pi$
$758$	0	0
$759$	−3612.89	−0.172779
$760$	0	0
$761$	−6947.99	−0.330965	−0.165483	−	0.986213i	$-0.552918\pi$
$761$	−6947.99	−0.330965	−0.165483	+	0.986213i	$0.552918\pi$
$762$	0	0
$763$	−8759.06	−0.415595
$764$	0	0
$765$	7482.27	0.353624
$766$	0	0
$767$	−42290.5	−1.99090
$768$	0	0
$769$	7189.91	0.337159	0.168579	−	0.985688i	$-0.446082\pi$
$769$	7189.91	0.337159	0.168579	+	0.985688i	$0.446082\pi$
$770$	0	0
$771$	−3053.95	−0.142653
$772$	0	0
$773$	−27591.1	−1.28381	−0.641904	−	0.766785i	$-0.721855\pi$
$773$	−27591.1	−1.28381	−0.641904	+	0.766785i	$0.721855\pi$
$774$	0	0
$775$	−113.430	−0.00525745
$776$	0	0
$777$	−2336.35	−0.107872
$778$	0	0
$779$	1975.32	0.0908512
$780$	0	0
$781$	−32914.1	−1.50801
$782$	0	0
$783$	−344.187	−0.0157091
$784$	0	0
$785$	6619.38	0.300963
$786$	0	0
$787$	−10322.4	−0.467541	−0.233770	−	0.972292i	$-0.575106\pi$
$787$	−10322.4	−0.467541	−0.233770	+	0.972292i	$0.575106\pi$
$788$	0	0
$789$	−2986.59	−0.134760
$790$	0	0
$791$	−209.859	−0.00943330
$792$	0	0
$793$	66318.5	2.96978
$794$	0	0
$795$	8705.76	0.388379
$796$	0	0
$797$	40205.9	1.78691	0.893455	−	0.449153i	$-0.148274\pi$
$797$	40205.9	1.78691	0.893455	+	0.449153i	$0.148274\pi$
$798$	0	0
$799$	47465.0	2.10162
$800$	0	0
$801$	−7007.23	−0.309099
$802$	0	0
$803$	48447.6	2.12911
$804$	0	0
$805$	2637.24	0.115466
$806$	0	0
$807$	10706.8	0.467037
$808$	0	0
$809$	−42375.9	−1.84160	−0.920802	−	0.390031i	$-0.872464\pi$
$809$	−42375.9	−1.84160	−0.920802	+	0.390031i	$0.872464\pi$
$810$	0	0
$811$	−7082.68	−0.306667	−0.153333	−	0.988175i	$-0.549001\pi$
$811$	−7082.68	−0.306667	−0.153333	+	0.988175i	$0.549001\pi$
$812$	0	0
$813$	−18882.8	−0.814573
$814$	0	0
$815$	−546.945	−0.0235075
$816$	0	0
$817$	8725.11	0.373627
$818$	0	0
$819$	7581.78	0.323478
$820$	0	0
$821$	18154.3	0.771728	0.385864	−	0.922556i	$-0.373903\pi$
$821$	18154.3	0.771728	0.385864	+	0.922556i	$0.373903\pi$
$822$	0	0
$823$	−10437.1	−0.442058	−0.221029	−	0.975267i	$-0.570941\pi$
$823$	−10437.1	−0.442058	−0.221029	+	0.975267i	$0.570941\pi$
$824$	0	0
$825$	1435.38	0.0605739
$826$	0	0
$827$	1486.57	0.0625068	0.0312534	−	0.999511i	$-0.490050\pi$
$827$	1486.57	0.0625068	0.0312534	+	0.999511i	$0.490050\pi$
$828$	0	0
$829$	−9987.68	−0.418440	−0.209220	−	0.977869i	$-0.567092\pi$
$829$	−9987.68	−0.418440	−0.209220	+	0.977869i	$0.567092\pi$
$830$	0	0
$831$	6790.95	0.283484
$832$	0	0
$833$	17727.6	0.737364
$834$	0	0
$835$	−21536.9	−0.892594
$836$	0	0
$837$	−335.159	−0.0138409
$838$	0	0
$839$	2835.68	0.116685	0.0583424	−	0.998297i	$-0.481418\pi$
$839$	2835.68	0.116685	0.0583424	+	0.998297i	$0.481418\pi$
$840$	0	0
$841$	−24226.5	−0.993337
$842$	0	0
$843$	1517.70	0.0620076
$844$	0	0
$845$	−43668.9	−1.77782
$846$	0	0
$847$	−15026.8	−0.609596
$848$	0	0
$849$	−21054.6	−0.851110
$850$	0	0
$851$	−1681.49	−0.0677330
$852$	0	0
$853$	17544.2	0.704223	0.352111	−	0.935958i	$-0.385464\pi$
$853$	17544.2	0.704223	0.352111	+	0.935958i	$0.385464\pi$
$854$	0	0
$855$	4920.32	0.196809
$856$	0	0
$857$	12827.7	0.511301	0.255651	−	0.966769i	$-0.417710\pi$
$857$	12827.7	0.511301	0.255651	+	0.966769i	$0.417710\pi$
$858$	0	0
$859$	35061.5	1.39265	0.696324	−	0.717728i	$-0.254818\pi$
$859$	35061.5	1.39265	0.696324	+	0.717728i	$0.254818\pi$
$860$	0	0
$861$	−1242.88	−0.0491953
$862$	0	0
$863$	4222.11	0.166538	0.0832691	−	0.996527i	$-0.473464\pi$
$863$	4222.11	0.166538	0.0832691	+	0.996527i	$0.473464\pi$
$864$	0	0
$865$	40868.2	1.60643
$866$	0	0
$867$	−3157.23	−0.123674
$868$	0	0
$869$	50448.4	1.96933
$870$	0	0
$871$	−35443.0	−1.37881
$872$	0	0
$873$	14432.1	0.559511
$874$	0	0
$875$	−15380.6	−0.594238
$876$	0	0
$877$	17535.0	0.675159	0.337579	−	0.941297i	$-0.390392\pi$
$877$	17535.0	0.675159	0.337579	+	0.941297i	$0.390392\pi$
$878$	0	0
$879$	28466.5	1.09232
$880$	0	0
$881$	9116.63	0.348634	0.174317	−	0.984690i	$-0.444228\pi$
$881$	9116.63	0.348634	0.174317	+	0.984690i	$0.444228\pi$
$882$	0	0
$883$	41050.4	1.56450	0.782251	−	0.622963i	$-0.214071\pi$
$883$	41050.4	1.56450	0.782251	+	0.622963i	$0.214071\pi$
$884$	0	0
$885$	17268.6	0.655907
$886$	0	0
$887$	−22519.5	−0.852461	−0.426230	−	0.904615i	$-0.640159\pi$
$887$	−22519.5	−0.852461	−0.426230	+	0.904615i	$0.640159\pi$
$888$	0	0
$889$	12178.3	0.459444
$890$	0	0
$891$	4241.22	0.159468
$892$	0	0
$893$	31212.9	1.16965
$894$	0	0
$895$	−33197.7	−1.23986
$896$	0	0
$897$	5456.66	0.203113
$898$	0	0
$899$	158.241	0.00587055
$900$	0	0
$901$	−20822.6	−0.769924
$902$	0	0
$903$	−5489.88	−0.202316
$904$	0	0
$905$	−50947.0	−1.87131
$906$	0	0
$907$	32037.9	1.17288	0.586439	−	0.809993i	$-0.300529\pi$
$907$	32037.9	1.17288	0.586439	+	0.809993i	$0.300529\pi$
$908$	0	0
$909$	11061.4	0.403612
$910$	0	0
$911$	12881.7	0.468485	0.234242	−	0.972178i	$-0.424739\pi$
$911$	12881.7	0.468485	0.234242	+	0.972178i	$0.424739\pi$
$912$	0	0
$913$	−6982.69	−0.253114
$914$	0	0
$915$	−27080.0	−0.978402
$916$	0	0
$917$	12299.8	0.442937
$918$	0	0
$919$	7311.92	0.262457	0.131229	−	0.991352i	$-0.458108\pi$
$919$	7311.92	0.262457	0.131229	+	0.991352i	$0.458108\pi$
$920$	0	0
$921$	13981.7	0.500232
$922$	0	0
$923$	49711.3	1.77277
$924$	0	0
$925$	668.047	0.0237462
$926$	0	0
$927$	−1935.59	−0.0685795
$928$	0	0
$929$	35059.5	1.23817	0.619087	−	0.785322i	$-0.287503\pi$
$929$	35059.5	1.23817	0.619087	+	0.785322i	$0.287503\pi$
$930$	0	0
$931$	11657.6	0.410379
$932$	0	0
$933$	13135.5	0.460919
$934$	0	0
$935$	43530.8	1.52258
$936$	0	0
$937$	25471.7	0.888072	0.444036	−	0.896009i	$-0.353546\pi$
$937$	25471.7	0.888072	0.444036	+	0.896009i	$0.353546\pi$
$938$	0	0
$939$	13332.5	0.463354
$940$	0	0
$941$	22426.5	0.776923	0.388461	−	0.921465i	$-0.373007\pi$
$941$	22426.5	0.776923	0.388461	+	0.921465i	$0.373007\pi$
$942$	0	0
$943$	−894.508	−0.0308899
$944$	0	0
$945$	−3095.89	−0.106571
$946$	0	0
$947$	36090.9	1.23843	0.619216	−	0.785220i	$-0.287450\pi$
$947$	36090.9	1.23843	0.619216	+	0.785220i	$0.287450\pi$
$948$	0	0
$949$	−73171.9	−2.50291
$950$	0	0
$951$	−23644.8	−0.806242
$952$	0	0
$953$	−16379.8	−0.556762	−0.278381	−	0.960471i	$-0.589798\pi$
$953$	−16379.8	−0.556762	−0.278381	+	0.960471i	$0.589798\pi$
$954$	0	0
$955$	−13919.2	−0.471639
$956$	0	0
$957$	−2002.43	−0.0676378
$958$	0	0
$959$	22935.8	0.772301
$960$	0	0
$961$	−29636.9	−0.994828
$962$	0	0
$963$	14279.2	0.477820
$964$	0	0
$965$	12612.7	0.420742
$966$	0	0
$967$	22686.2	0.754434	0.377217	−	0.926125i	$-0.376881\pi$
$967$	22686.2	0.754434	0.377217	+	0.926125i	$0.376881\pi$
$968$	0	0
$969$	−11768.5	−0.390154
$970$	0	0
$971$	−53773.7	−1.77722	−0.888610	−	0.458663i	$-0.848328\pi$
$971$	−53773.7	−1.77722	−0.888610	+	0.458663i	$0.848328\pi$
$972$	0	0
$973$	−25482.4	−0.839597
$974$	0	0
$975$	−2167.90	−0.0712086
$976$	0	0
$977$	−20398.6	−0.667972	−0.333986	−	0.942578i	$-0.608394\pi$
$977$	−20398.6	−0.667972	−0.333986	+	0.942578i	$0.608394\pi$
$978$	0	0
$979$	−40767.0	−1.33087
$980$	0	0
$981$	7400.30	0.240850
$982$	0	0
$983$	12118.6	0.393208	0.196604	−	0.980483i	$-0.437009\pi$
$983$	12118.6	0.393208	0.196604	+	0.980483i	$0.437009\pi$
$984$	0	0
$985$	−49074.8	−1.58746
$986$	0	0
$987$	−19639.3	−0.633359
$988$	0	0
$989$	−3951.10	−0.127035
$990$	0	0
$991$	26885.7	0.861809	0.430904	−	0.902398i	$-0.358195\pi$
$991$	26885.7	0.861809	0.430904	+	0.902398i	$0.358195\pi$
$992$	0	0
$993$	−381.181	−0.0121817
$994$	0	0
$995$	−5099.10	−0.162465
$996$	0	0
$997$	21022.7	0.667797	0.333899	−	0.942609i	$-0.391636\pi$
$997$	21022.7	0.667797	0.333899	+	0.942609i	$0.391636\pi$
$998$	0	0
$999$	1973.93	0.0625148

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.4.a.h.1.2		2
4.3	odd	2		69.4.a.a.1.1	✓	2
12.11	even	2		207.4.a.c.1.2		2
20.19	odd	2		1725.4.a.n.1.2		2
92.91	even	2		1587.4.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.a.a.1.1	✓	2	4.3	odd	2
207.4.a.c.1.2		2	12.11	even	2
1104.4.a.h.1.2		2	1.1	even	1	trivial
1587.4.a.b.1.1		2	92.91	even	2
1725.4.a.n.1.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -10.7639 q^{5} -10.6525 q^{7} +9.00000 q^{9} +52.3607 q^{11} -79.0820 q^{13} +32.2918 q^{15} -77.2361 q^{17} -50.7902 q^{19} +31.9574 q^{21} +23.0000 q^{23} -9.13777 q^{25} -27.0000 q^{27} +12.7477 q^{29} +12.4133 q^{31} -157.082 q^{33} +114.663 q^{35} -73.1084 q^{37} +237.246 q^{39} -38.8916 q^{41} -171.787 q^{43} -96.8754 q^{45} -614.545 q^{47} -229.525 q^{49} +231.708 q^{51} +269.597 q^{53} -563.607 q^{55} +152.371 q^{57} +534.768 q^{59} -838.604 q^{61} -95.8723 q^{63} +851.234 q^{65} +448.180 q^{67} -69.0000 q^{69} -628.604 q^{71} +925.266 q^{73} +27.4133 q^{75} -557.771 q^{77} +963.479 q^{79} +81.0000 q^{81} -133.358 q^{83} +831.364 q^{85} -38.2430 q^{87} -778.581 q^{89} +842.420 q^{91} -37.2399 q^{93} +546.703 q^{95} +1603.57 q^{97} +471.246 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 26 q^{5} + 10 q^{7} + 18 q^{9} + 60 q^{11} - 24 q^{13} + 78 q^{15} - 150 q^{17} + 46 q^{19} - 30 q^{21} + 46 q^{23} + 98 q^{25} - 54 q^{27} - 216 q^{29} - 324 q^{31} - 180 q^{33} - 200 q^{35}+ \cdots + 540 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 26 * q^5 + 10 * q^7 + 18 * q^9 + 60 * q^11 - 24 * q^13 + 78 * q^15 - 150 * q^17 + 46 * q^19 - 30 * q^21 + 46 * q^23 + 98 * q^25 - 54 * q^27 - 216 * q^29 - 324 * q^31 - 180 * q^33 - 200 * q^35 + 140 * q^37 + 72 * q^39 - 364 * q^41 + 126 * q^43 - 234 * q^45 - 120 * q^47 - 146 * q^49 + 450 * q^51 + 490 * q^53 - 680 * q^55 - 138 * q^57 + 32 * q^59 - 908 * q^61 + 90 * q^63 + 12 * q^65 + 874 * q^67 - 138 * q^69 - 488 * q^71 + 652 * q^73 - 294 * q^75 - 400 * q^77 + 1198 * q^79 + 162 * q^81 + 100 * q^83 + 1940 * q^85 + 648 * q^87 + 102 * q^89 + 1980 * q^91 + 972 * q^93 - 928 * q^95 + 1624 * q^97 + 540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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